Central Difference Method Of Solving The Conservation Laws And Convection Diffusion Equation

In recent years, a tremendous amount of research was done in developing Computational Fluid Dynamic (CFD). Finite difference schemes are one of important methods. For the need of neither Riemann-solvers nor characteristic decomposition, central schemes have attracted a lot of attention.In this paper, a new fifth-order compact central weighted ENO scheme was presented based on central weighted ENO and third-order compact central weighted ENO scheme. The scheme has much higher accuracy and no oscillations. It not only keeps the virtue of the central scheme, requiring no Riemann solvers, no projection along characteristic direction, and no flux splitting, but also needs less point at the same accuracy. The scheme is based on an extremely compact five-point stencil. Several numerical tests demonstrate those virtues.In fact, central schemes are based on cell-average u_jⁿ at time tⁿ and then project on staggered cell-average u_(j+1)/2ⁿ⁺¹ at next time tⁿ⁺¹. And it is difficult to program andto deal with boundary conditions. In this paper, non-staggered central scheme was used to solve this problem and demonstrated no oscillations and high-resolution properties in hyperbolic conservation laws and convection-diffusion equations. It means that non-staggered central scheme is better than central scheme.